Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T05:50:33.718Z Has data issue: false hasContentIssue false
  • English
  • Français

The higher Stasheff-Tamari posets

Part of: Ordered sets

Published online by Cambridge University Press:  26 February 2010

Paul H. Edelman  and
Victor Reiner
Paul H. Edelman
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A.
Victor Reiner
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A.
Get access

Abstract

This paper studies higher dimensional analogues of the Tamari lattice on triangulations of a convex n-gon, by placing a partial order on the triangulations of a cyclic d-polytope. Our principal results are that in dimension d≤3, these posets are lattices whose intervals have the homotopy type of a sphere or ball, and in dimension d≤5, all triangulations of a cyclic d-polytope are connected by bistellar operations.

MSC classification

Secondary: 06A07: Combinatorics of partially ordered sets
Type
Research Article
Information
Mathematika , Volume 43 , Issue 1 , June 1996 , pp. 127 - 154
Copyright
Copyright © University College London 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ba.Balinski, M.. On the graph structure of convex polyhedra in n-space. Pac. J. Math., 11 (1961), 431–434.CrossRefGoogle Scholar
Bj.Bjőrner, A.. Homotopy type of posets and lattice complementation. J. Combin. Th. Ser. A, 30 (1981), 90–100.CrossRefGoogle Scholar
BFS.Billera, L. J., Filliman, P. and Sturmfels, B.. Constructions and complexity of secondary polytopes. Adv. Math. 83 (1990), 155–179.CrossRefGoogle Scholar
BGS.Billera, L. J., Gelfand, I. M. and Sturmfels, B.. Duality and minors of secondary polyhedra. J. Combin. Th. Ser. B, 57 (1993), 258–268.CrossRefGoogle Scholar
BKS.Billera, L. J., Kapranov, M. M. and Sturmfels, B.. Cellular strings on polytopes. Proc. Amer. Math. Soc, 122 (1994), 549–555.CrossRefGoogle Scholar
BP.Billera, L. J. and Provan, S.. A decomposition property for simplicial complexes and its relation to diameters and shellings. Ann. of the New York Academy of Sciences, 319 (1979), 82–85.CrossRefGoogle Scholar
BS. Billera, L. J. and Sturmfels, B.. Fiber polytopes. Ann. Math., 135 (1992), 527–549.CrossRefGoogle Scholar
BW.Bjőrner, A. and Wachs, M.. Shellable nonpure complexes and posets, II. Preprint, 1994.Google Scholar
ER.Edelman, P. H. and Reiner, V.. Free arrangements and rhombic tilings. Disc. & Comp. Geom. (to appear).Google Scholar
Ge.Geyer, W.. On Tamari lattices. Disc. Math., 133 (1994), 99–122.CrossRefGoogle Scholar
GKZ1.Gelfand, I. M., Kapranov, M. M. and Zelevinsky, A. V.. Discriminants of polynomials in several variables and triangulations of Newton polytopes. Leningrad Math. J., 2 (1991), 449–505.Google Scholar
GKZ2.Gelfand, I. M., Kapranov, M. M. and Zelevinsky, A. V.. Discriminants, resultants and multidimensional determinants (Birkhäuser, Boston, 1994).CrossRefGoogle Scholar
Gr.Grünbaum, B.. Convex polytopes (Wiley and Sons, New York, 1967).Google Scholar
Ha.Haiman, M.. Constructing the associahedron. Manuscript.Google Scholar
HT.Huang, S. and Tamari, D.. Problems of associativity: A simple proof for the lattice property of systems ordered by a semi-associative law. J. Combin. Theory Ser. A, 13 (1972), 7–13.CrossRefGoogle Scholar
KV.Kapranov, M. M. and Voevodsky, V. A.. Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (list of results). Cah. de Top. et Geom. Diff. Categor., 32 (1991), 11–27.Google Scholar
Le.Lee, C.. The associahedron and triangulations of the n-gon. Europ. J. Combinatorics, 10 (1989), 551–560.CrossRefGoogle Scholar
deL. Jesùs de Loera. Computing regular triangulations of point configurations. Preprint Manual for PUNTOS, 1994.Google Scholar
MS. Manin, Y. I. and Schechtman, V. V.. Arrangements of hyperplanes, higher braid groups, and higher Bruhat orders. Adv. Stud, in Pure Math., 17 (1989), 289–308.CrossRefGoogle Scholar
Pac.Pachner, U.. P.L. homeomorphic manifolds are equivalent by elementary shellings. Europ. J. Combinatorics, 12 (1991), 129–145.CrossRefGoogle Scholar
Pall. Pallo, J. M.. On the rotation distance in the lattice of binary trees. Inf. Process. Let., 25 (1987), 369–373.CrossRefGoogle Scholar
Pal2.Pallo, J. M.. Some properties of the rotation lattice of binary trees. The Comp. Jour., 31 (1987), 564–565.Google Scholar
Pal3.Pallo, J. M.. A distance metric on binary trees using lattice-theoretic measures. Inf. Proc. Let., 34 (1990), 113–116.,CrossRefGoogle Scholar
Pal4.Pallo, J. M.. An algorithm to compute the Mobius function of the rotation lattice of binary trees. Theor. Informatics and Applic, 27 (1993), 341–348.CrossRefGoogle Scholar
Sa.Sagan, B.. A generalization of Rota's NBC theorem. Adv. Math., 1ll (1995), 195–207.CrossRefGoogle Scholar
St.Stasheff, J. D.. Homotopy associativity of H-spaces. Trans. Amer. Math. Soc, 108 (1963), 275–292.Google Scholar
STT.Sleator, D., Tarjan, R. and Thurston, W.. Rotation distance, triangulations, and hyperbolic geometry. J Amer. Math. Soc, 1 (1988), 647–681.CrossRefGoogle Scholar
Ta.Tamari, D.. The algebra of bracketings and their enumeration. Nieuw Arch. Wisk., 10 (1962), 131–146.Google Scholar
Zi.Ziegler, G.. Higher Bruhat orders and cyclic hyperplane arrangements. Topology, 32 (1993), 259–279.CrossRefGoogle Scholar